Theoretical Study On The Minimum Structure Of High-order Harmonic Emission

High-order harmonic generation（HHG）is a type of high-energy electromagnetic radiation generated during the interaction between high-intensity laser pulses and atoms or molecules.Its wavelength can be extended from ultraviolet to the soft X-ray regions.Since there is a"platform"structure in the high-order harmonic emission spectrum that changes little in intensity as the frequency increases,it is used to generate ultrashort pulses on the attosecond time scale to detect ultrafast electron motion.The mechanism of HHG is that the ionized electrons return to the parent ion,recombine with the parent nuclei,and radiate high-energy photons.Therefore,the instantaneous structural characteristics of atoms and molecules are encoded into the harmonic spectrum.The return of ionized electrons to the parent ion can lead to harmonic interference from multiple atomic centers,resulting in minimum in the HHG spectrum.The minima of HHG are related to the molecular bond length,so it can be used to detect instantaneous bond length changes in simple molecules.Due to the complexity of molecular structures and the laser field during HHG,there may be significant differences between the observed harmonic minima and the results predicted by the two-centre interference model.To understand the reasons behind the formation of harmonic minima,this paper investigates the effects of the driving laser field and multiple electronic orbitals of the molecule on the minimum using numerical solutions of the time-dependent Schr（?）dinger equation（TDSE）and time-dependent density-functional theory（TDDFT）approaches.The main research contents are as follows:First,by numerically solving the TDSE,the harmonic generation process of hydrogen molecular ion（H₂⁺）driven by mid-infrared（MIR）laser fields was investigated.It is observed that under conditions of increasing the distance of molecular nuclei,multiple minima exist in the harmonic spectra.As the wavelength of the driving laser decreases and the intensity increases,these minima gradually shift towards higher energy regions.This is inconsistent with the predictions of the previous two-center interference model for the positions of the minima.To comprehend the underlying causes of harmonic minima variations at larger nuclei distances,we investigated the mechanism of harmonic emission by utilizing the time-frequency analysis behavior and analyzing different central emission schemes.On the basis of the two-center interference model,the influence of the driving laser is added,the additional potential energy brought by the laser electric field is introduced,and a new revised formula that can accurately describe the position of the high-order harmonic minimum obtained by numerical simulation is proposed.Subsequently,we further investigate the effects of nuclei motion and the pulse width of the driving laser on the positions of harmonic minima.Through analysis of the time-frequency behavior,we find that our formula can be effectively applied under these conditions.By increasing the pulse duration in the case of single color field,the averaging effect due to multiple emissions results in the minima of the harmonic emission spectra being smoothed out,posing significant challenges for its observation in experiments.In order to solve this problem,we propose a two-color field scheme to study the harmonic emission spectra of molecules driven by longer duration laser pulses.Minima can be clearly observed in the harmonic spectra,and the positions of these minima can be well explained by our revised formula.On this basis,the HHG of H₂⁺under the action of a two-color laser electric field is studied.It is found that this scheme can be used to effectively adjust the position of the minimum,and then an ultra-short pulse of 71 as was obtained by synthesizing harmonics,which provides a new idea for generating isolated attosecond pulses.Our conclusions are tested by numerically solving two-dimensional TDSE.Furthermore,by altering the aligment angle between the driving laser and the molecular axis,it is observed that the positions of minima in the harmonic emission spectra varied significantly.However,these variations could still be described using our revised formula incorporating the effect of the laser field.Secondly,the time-dependent density functional theory is employed to study the harmonic emission minima in symmetric and asymmetric molecules.The diatomic molecules nitrogen（N₂）and oxygen（O₂）,as well as the triatomic molecules carbon dioxide（CO₂）and carbon disulfide（CS₂）are selected.The driving laser pulse is a linearly polarized ultra-short pulse,with the polarization direction of the laser field perpendicular to the molecular axis.It is found that the harmonic emission spectrum of N₂does not show a minimum structure under the same laser conditions,whereas a minimum is observed in the harmonic spectrum of the O₂.Through time-frequency analysis,it is found that the HHG of N₂is predominantly from the HOMO-1 b orbital,while the minimum in the harmonic emission spectrum of O₂originate from the harmonic interference between its HOMO and HOMO-1 orbitals.For the triatomic molecule CS₂,minimum structure is observed,while this feature is not seen in the harmonic spectrum of CO₂.Through analysis of the time-dependent electron probabilities as well as the orbital transition dipole,it is found that the HHG of CO₂mainly originated from the HOMO-2 b orbital,while the harmonic contribution of CS₂come from the interference of its HOMO b and HOMO-1 b orbitals.Furthermore,the HHG of the asymmetric molecule nitrous oxide（N₂O）is studied under conditions of linear polarization of the driving laser with the polarization direction perpendicular to the molecular axis.It is demonstrated that a minimum exists in the plateau region of the N₂O harmonic spectrum in the direction of laser polarization.Through analysis of the time-dependent probabilities of different electronic orbitals,time-dependent of wave packets,and transition dipole of different orbitals,it is found that this minimum is the result of harmonic interference from the HOMO a,HOMO-1,and HOMO-3 a orbitals.Based on this,the effects of the driving laser intensity and wavelength on this structure are discussed,and it is find that such a minimum can be observed within a certain range of laser parameters.Thirdly,the mechanism of modulating harmonic efficiency is investigated.We studied the motion behaviour of the electrons of the system under the influence of a weak static electric field orthogonal to a linearly polarized infrared（IR）pulse.It is found that the orthogonal weaker electrostatic field leads to a change in the motion of the ionized electrons in the direction perpendicular to the laser polarisation direction,so that the distribution of the electron wave packets in different spatial partitions is no longer symmetric,thus modulating the harmonic efficiency.Through this scheme,harmonic electric fields with high ellipticity can be synthesized.After it is clear that the motion of ionized electrons in the orthogonal laser polarization direction could be controlled by adjusting the electrostatic field amplitude,we further study how to control the harmonic minimum by regulating the trajectory of ionized electrons in the orthogonal laser polarization direction.The target particle is an asymmetric carbonyl sulfide（OCS）molecule with a large permanent dipole moment.When driven by near-infrared pulses,as sulfur has a weaker electronegativity,ionization mainly occurs from the sulfur nuclei region,and the returning electron wave packet also predominantly returns to the sulfur nuclei region.Consequently,the harmonic spectrum in the polarization direction of the near-infrared pulse dose not exhibit minimum structure.However,when driven by both near-infrared pulses and a weak orthogonal terahertz（THz）field,minimum is observed in the harmonic spectrum along the polarization direction of the near-infrared laser.Analysis indicated that under the influence of the THz field,electrons can move perpendicularly to the polarization direction of the near-infrared pulse.This gives the electrons,after ionizing from the sulphur side of the molecule,the opportunity to recombine with the sulphur side and the oxygen side,respectively,thus generating harmonic emission near the two different nuclei regions and their interference leading to a minimum.Additionally,we find that the combination of fields driving OCS could synthesize near-circularly polarized attosecond pulses,and by altering the carrier-envelope phase（CEP）of the near-infrared laser pulse,the ellipticity of the attosecond pulses could be controlled.In addition,we simulate the HHG of OCS molecules driven by mid-infrared laser pulses.By controlling the ionization electron trajectory perpendicular to the laser polarization direction,we achieved manipulation of the minimum structure in the harmonic spectrum.It is observed that the harmonic spectra in both the parallel and perpendicular（molecular axis direction）laser polarization directions showed minima.Through the time-dependent evolution analysis of the wave packet,it is found that in MIR pulse polarization,the ionized electron wave packet begins localized and then undergoes dispersion.The large spatial dispersion of ionized electrons due to MIR pulse driving,combined with the effect of the permanent dipole moment,causes the ionized electrons to recombine with the molecular ion across the entire molecular region upon returning,resulting in the destructive interference and formation of minimum.Along the molecular axis direction,due to the large permanent dipole of OCS,the ionized electrons returning to the nuclei region were deflected along the molecular axis,causing interference and minima in the harmonics from different centers in this polarization direction.These research results provide new directions for in-depth understanding and regulation of harmonic minima.